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Thermal response of composite cylinders
INT. 00MM. HEAT MASS TRANSFER 0735-1933/87 $3.00 + .00 Vol. 14, pp. 417-427, 1987 ©Pergamon Journals Ltd. Printed in t h e U n i t e d S t a t e s
THERMAL RESPONSE OF COMPOSITE CYLINDERS
R. K. Chohan Nuclear Engineering Department University of Tennessee, Knoxville
(C~,,L~nicated by J.P. Hartnett and W.J. Minkowycz)
ABSTRACT Composite cylinders are not uncommon in present-day technology. Insulated plpes~ nuclear fuel rods, optical fibers and heat pipes may be dited as examples. A complex composite cylinder is an industrial thermometer of the immersion type. Such thermometers contain at least three components: an outer sheath usually made of stainless steel, a ceramic insulant such as magnesia, and the sensing element which is usually a thermocouple junction or a resistance temperature detector. Models of such cylinders have been dev~loped and are briefly described. The thermal response of these cylinders depends on a number of factors. These include physical properties and dimensions which can be influenced by manufacturing tolerances. Some theoretical and experimental results are presented to exemplify this. Simple representation of complex cylinders is briefly discussed, Some results of thermal processing of food cans are also presented.
Introduction Composite cylinders are not u n c o m o n in present day technology. Besides insulated pipes, the following may be cited as examples: nuclear fuel rods, optical fibers, heat pipes and food cans. A somewhat less obvious example is an industrial thermometer. The components of a composite cylinder attempt to provide one or more functions. The cladding of an optical fiber has a refractive index lower than the core which results in the confinement of a light ray in the core,
*Present address:
Dept. of Chemical Engineering, Technlon City, Haifa 32000, Israel. 417
418
R.K. Chohan
Vol. 14, No. 4
Food cans provide a containing medium as well as a means of preserving A classification the possible designs. (I)
of composite cylinders may assist in conceptualising Three classes of composite cylinders can be identified:
Composite cylinder with annular regions. is composed of different materials
In this class, the cylinder
in annular regions.
One or more of
these annular regions can itself have a composite characteristic perhaps a complex geometric (II)
:food,
form.
Composite cylinders with multiple filaments running longitudinally
with
(or smaller cylinders)
along the cylinder.
(III) Composite cylinders with disperse medium.
In this class, a geometrically
dominant disperse and continuous phase can be identified. Examples of class thermometers.
(I) are insulated pipes and some forms of industrial
Thermocouple
cable leads and some types of optical fiber cables
belong to class (II). Food cans are examples of class
(III).
Not all com-
posite cylinders can neatly be classified as one of the above. some forms of industrial thermometers
Examples are
and optical fiber cables.
This paper attempts to present some of the author's experiences transfer in composite cylinders. belong to class
These include industrial
(I) or a cross between class
of the following discussion
is on the former.
specifically
for industrial thermometers
thermometers,
which
(I) and (II), and food cans. Most Some theoretical and experiment-
al results of the thermal response are presented.
just sensors)
in heat
Though these are obtained
(also called temperature
and food cans, such results should find relevance
sensors or
to other
composites. Factors Affecting Industrial
the Thermal Response
thermometers are composite cylinders which belong to class
(I) or have characteristics
of class (I) and class
of these are shown in figures 1 and 2.
(If). Examples of designs
Further examples are presented
[1,2].
A typical sensor has at least the following components:
sheath,
(ii) an inner layer
(platinum element,
(a ceramic insulant),
in
(i) an outer
(iii) the sensing element
thermistor bead or thermocouple junction which normally
occupies a negligible volume),(iv)
core
present such as in figure 2. Undesired may be introduced during manufacture,
(or mandrel), Additional layers may be components
(such as air or cracks)
handling or use.
Vol. 14, NO. 4
THEI~9~RESPONSE OF O O M P ( ~ I T E ~
419
Processes of interest in composite cylinder analysis and design fall into two categories (3): (a) Short-term processes such as heat transfer within cylinder and between cylinder surface and the environment, flow of fluid surrounding cylinder, vibration, etc. (b) Long-term processes such as corrosion, fatigue, crack and crack propagation, scale formation, erosion, wear, etc. Design considerations such as system function and reliability involves both the above categories. The thermal response of composite cylinders may be influenced by both these categories. The thermal response of temperature sensors is affected by a number of factors [1,5]. Generally, the major determinants of the thermal response are the dimensions and geometrical form of the constituents, physical properties, temperature and pressure. Because of the relatively small outer diameters of average temperature sensors, manufacturing tolerances can be expected to have an influence on the thermal response. It has been found that some of the manufacturing tolerances may have a significant, possibly a dominant, influence [1,2,4].
Some results
presented here exemplify this. Thermal response of systems may be represented as a time history to a specified perturbation or by a single parameter, T .
The latter is usually
taken to be the time required for the response to reach 63.2 percent of the total change following a step perturbation. This defines the time constant which however, strictly, applies to a first-order lumped-parameter system whilst temperature sensors are higher order systems. In fact they are distributed-parameter systems. However, the time constant as a measure of thermal response is widely used. Two kinds of perturbations are of interest for composite cylinders: (a) heat input from the environment, (b) heat input to a location inside the cylinder. Industrial thermometry experimentation and modelling simulates these perturbations as follows: (i)
Plunge test (abbreviation PT) - where the thermometer is plunged into hot water from ambient air,
420
R.K. Chohan
(ii) Electrical
Vol. 14, No. 4
transient test (abbreviated ET) - where the thermometer
is held at a constant temperature water bath at room temperature and the current input into the thermometer approximately
is a step change from
1-5 mA to about 30-50 mA. Substantial
internal
(self) heating results. Most of the results presented below pertain to the first of these two perturbations.
Modeling Attempts of modeling of industrial thermometers have been made. Both distributed
and lumped parameter models have been developed
[5,6]. Some
results are presented here. The distributed-parameters
representation
of a composite medium is
desirable because of the accurate representation process.
of the heat transfer
The finite element method is particularly
suited to composites
because of the relative ease with which differing physical properties be accommodated
In the distributed models,
it was assumed that only radial heat
transfer is present and that the physical properties are constant. temperature
sensor together with a thermowell,
into the following regions: (iv) air gap (inner), sensing element decomposed
can
[14].
(i) thermowell,
(v) aluminum cover,
if present,
(ii) air gap,
The
can be decomposed (iii) sheath,
(vi) ceramic former containing
the
(here a platinum coil). The last region was further
into (a) an outer
wire (assumed to be massless)
(annular) ceramic, imbedded in it,
(b) air gap with sensing
(c) inner cylindrical core.
Other sensor types can be similarly decomposed. Each of these regions four-noded axisymmetric kp and heat capacity, if the thermowell
c.
(steel well, air gap, etc.) was divided into one
line element with appropriate At the thermowell surface
is not present),
appropriate value of
h
thermal conductivity,
(or at the sheath surface
a surface element was added with an
(the surface heat transfer coefficient).
The result-
ing model had 9 elements with 25 nodes, with node 5 from the cylinder center being the temperature of the sensing element. The model was simulated using a finite element package on a minicomputer [5].
Typical outer (inner) diameters of well and sheath were i0 m~ (6,2 mm)
and 6.0 mm (5.07 mm); details can be found in (5) and (8). Figure 3 shows
Vol. 14, No. 4
THEI~MAL RESPONSE (~ CQMIK~ITE CYLINDERS
~
Sheath
~
insulant
Fig. i. A typical sensor for Temperature measurement with possible locations of sensing element: (A) insulated junction, (B) grounded junction, (C) resistance detector.
well heath luminium
cover
ceramic
~ i r /
-sensing
element
gaps Fig. 2. An industrial resistance temperature sensor in a well (cross-sectional view).
0
.
0
Z Ttm
II (secaeds)
~ 14 10
Fig. 3. Comparison between distributed model and experiment.
ZO
421
422
R.K. Chohan
Vol. 14, No. 4
typical model simulation results and experimental data for both types of perturbations described in the last section. The results are for an industrial thermometer
(shown schematically in figure 2). Agreement is seen
to be a good showing the validity of the distributed parameter model. Note that there are two sets of experiments which are distinct and validity is tested with respect to both. The distributed parameter models require a relatively large computer time and memory. Lumped and reduced-order models are desirable in this respect.
It has been found that valid reduced-order models can be constructed
for a class of thermometers shown in figure (2) [7,5].
These validated
models represent the sensor as a third-order lumped parameter system and the sensor-well combination as a fourth-order system. That is, a sensor response can be described by:
T(t)=l+ale-Plt+a2e-P2t+a3e-P3t where the a i , Pi
are constants, T
(1)
the sensor response, t
Two-dimensional lumped models have also been developed
is time. [6].
These
allow radial and axial heat transfer. Except for short cylinders, radial heat transfer is the dominant mode. Two-dimensional models allow study of short cylinders and steam loss errors.
Manufacturin $ Tolerances Manufacturing tolerances appear as dimensional tolerances and physical property variations. Two important tolerances in industrial thermometry are air gap dimensions
(see figure 2) and the porosity of the insulant
(such
as magnesia and alumina). An industrial sensor in a thermowell
(figure 2) was simulated using
the finite element model (described in the last section).
Manufacturing
tolerances in air gap dimensions usually involve very small dimensional changes about a nominal value.
Figure 4 is a typical set of results showing
the effects of dimensional tolerances in the air gaps. For the relatively small changes involved, an almost linear variation in the response time is obtained. A significant variation in the time constants
(50% or more) can
be expected in the manufacture of sensors with thermowells. This sort of variation has been observed in the laboratory 13].
VOl. 14, NO. 4
T}IER~%L RESPONSE (F OOMPOSITE CYLINDERS
Experimental different
results have been obtained recently on the response of
sensors in five liquids.
of manufacturing
423
Some of the results exhibit the effects
tolerances not involving air gaps. These are shown in
figure 5 for two sensors in carbon tetrachloride.
Both the sensors are of
nominal outer diameter 6.20 mm, same make, type and manufacturer. significant difference difference
The
in the thermal response is probably due to the
in the insulant porosity.
First-Order Representation
of a Composite Cylinder
Composite cylinders are distributed parameter systems. Therefore, complete representation
a
of its thermal response would require an infinite
number of modes. As stated earlier, valid third-order lumped models could be constructed
for sensors. Earlier experiences
(5) appear to suggest that
there is one dominant mode, i.e. a sensor may approximately be described by a first-order representation. homogeneous
cylinder,
Simple analysis,
assuming a lumped or
suggests that the time constant may be expressed as [9]: C2 = cI +~-
where
CI
and
C2
(2)
are dependent only on the sensor internal dimensions
and physical properties.
The effects of the external fluid are contained
entirely in the heat transfer coefficient,
h .
Recent experiences also support the approximate validity of equation
(2) [6,10j.
Two composite cylinders of the type shown in figure i
of outer diameters 3 mm and 6 mm with sheath thickness 0.5 ~n and length 20 mm~ were simulated using a two-dimensional model described
in
6 .
The
insulant was alumina. The average response at the center of the cylinder was determined as a function of the heat transfer coefficient.
The results are
presented in figure 6.
i/h
suggesting
A linear relation between
the validity of (2).
T
and
is seen
It should be pointed out that this does
not mean that a first-order representation would agree with the entire time history as shown in figure 3. certain thermocouples,
Other composite cylinders,
were also simulated.
representing
Results were similar to those in
figure 6, again suggesting the approximate validity of expression
(2).
Food Cans Food cans belong to class
(III) composites.
the low length to diameter ratio.
They are characterised by
Significant axial, besides radial, heat
424
R.K. Chohan
Vol.
70
60
50 f f~
40 / /
o
/
30 E
/
J
/ 20
h
f6
26
air gap width (mm x 10 .2 ) Fig. Effect of air gap widths of a temperature detector. and broken line- inner
4. on the thermal response Full line-outer air gap, air gap (see Fig. 2).
8.0
6.0 v
0
~
4.0
$ I,--
2.0
OtS
1:0
Fluid Velocity (m/s) Fig. 5. Thermal response of two sensors of the same type, make, vendor model, and nominal dimension in carbon tetrachloride.
14, No. 4
VOI. 14, No. 4
T ~
~-_~SE
OF C X ~ I T ~
C
~
<=. : " C3 U')
V- l= ~
x~ x
%.60
x
o'.os
6.~5
o'.,o
(M--2.C/W)
1/'H
"~.2o
"lO"
Fig. 6. Time constant-Heat transfer Coefficient relation for a 3 mm (crosses) and 6 mm (squares) sensor.
:--..-;-~'~-- ~
2,0
e • m
' ~
#
/ 4
I
/
^F
C~R
• f v Z00
[
£
;
;
O~ CRUS~ CO~N cAs
~rr~
' I •
. . . . . . . .
SURFACE
[~',
.
.
.
.
...... x,.,
SURFACE OF W~OLECORN CAN * . . . . •
•
..."
p,___._-o
~
I,;
is
Tfeem ( l l n u ~ e s )
Fig. 7. Surface and interior temperatures of crushed and whole corn cans in a still vertical retort.
425
426
R.K. Chohan
Vol. 14, No. 4
transfer may be expected. A number of experiments were performed The cans were steam heated in a vertical and surface measurements
!111,12].
still retort under pressure.
were obtained and are described
Internal
in [11,12]. Complex
temperature histories are possible at internal locations of the cans. Some comparisons
between experimental data are presented
in figure 7.
~his shows the results from two cans, one with 350 g. crushed corn and the other with 350 g. whole corn. Both cans were subjected to similar perturbation (steam heating to 250°F from about 70°F in the retort at about 2 atmosphere absolute pressure). temperature
Observations
can be stated as follows:
(a) the surface
of the whole corn responds faster than that of the crushed corn;
(b) the bottom inside temperature history exhibits a spike for crushed corn, which is absent for whole corn; can was
(c) the center temperature of the whole corn
(not shown in figure 7) much faster than that of the crushed corn.
Other results were obtained for cans containing potato chips in brine and starch solutions. (gelatinisation)
The latter were subject to phase transformation
[12]. This complicates
the heat transfer considerably.
Conclusion Some experience Finite-element
in heat transfer in composite cylinders was presented.
distributed
and lumped models have been developed and some
results were briefly discussed. ing tolerances.
Temperature
sensors are subject to manufactur-
These may have a significant effect on their thermal response.
Large temperature changes may result in physical property variations through its temperature dependence and also through geometrical expansion).
(differential
Model simulations can shed more light on thermometer behaviour
subjected to significant on food
changes
temperature changes
[13]. Some experimental
cans in a vertical still retort were presented.
results
These exemplify the
complex temperature histories that are possible in such cylinders.
References !.
T.W. Kerlin and R.K. Chohan, Chapter 15 in "Handbook of Applications of Measuring Instruments-Electrical Instruments", ed. R.L. Moore, Wiley, to be published (1986).
2.
R.K. Chohan and T.W. Kerlin, Paper 20 in the Proceedings of the Industrial Temperature Measurement Conf., Knoxville, TN (September 1984).
3.
R.K. Chohan and F. Abdullah, "Mathematical Modelling of Thermal Transducers' Progress report, Dept. of Systems Science, The City University, London, (August 1980).
Vol. 14, No. 4
THERMAL R E S P O N S E ~ O O M P O S I T E C Y L I N D E R S
427
4.
RoK, Chohan, "Industrial Thermometry-Dynamic Responsep Usage Environment and Sensor Location"p Submitted A.S.M.E. (1986).
5.
R.K, Chohan, Ph.D. Thesisp The City University, London (1983).
6.
R.K. Chohan, M. Natour and T.W. Kerlin, Paper ii in the Proceedings of the Industrial Temperature Measurement Conference, Knoxville, TN (1984).
7.
R.K. Chohan and F. Abdullah, "Understanding the Dynamic Response of Industrial Temperature Sensors- an Approach Based on Mathematical Models , presented at Transdueer/Tempcon Conf., Earls Court, London,(1983).
8.
R.K. Chohan, F. Abdullah and L. Finkelstein, "Mathematical Modelling of Industrial Thermometers", to be published in Trans. Inst. of Measurement and Control, (1986).
9.
T.W. Kerlin and R. Shepard, "Industrial Temperature Measurement", (1982).
ISA
i0.
T.W. Kerlin i R.K. Chohan and M. Natour, "Effects of Fluid Physical Properties and Flow on the Response of Temperature Sensors", to be submitted (1986).
ii.
J. Mount and R.K. Chohan, "Thermal Processing of Food Cans in a Still Retort", in preparation (1986).
12.
R.K. Chohan and J. Mount, "Simultaneous Surface and Internal Temperature Measurements in Food Cans in Retorts",in preparation (1985).
13.
R.K. Chohan, "The Response of Temperature Sensors", in progress (1986).
14.
L.J. Segerlind, "Applied Finite Element Analysis", Wiley, New York (1976).
THERMAL RESPONSE OF COMPOSITE CYLINDERS
R. K. Chohan Nuclear Engineering Department University of Tennessee, Knoxville
(C~,,L~nicated by J.P. Hartnett and W.J. Minkowycz)
ABSTRACT Composite cylinders are not uncommon in present-day technology. Insulated plpes~ nuclear fuel rods, optical fibers and heat pipes may be dited as examples. A complex composite cylinder is an industrial thermometer of the immersion type. Such thermometers contain at least three components: an outer sheath usually made of stainless steel, a ceramic insulant such as magnesia, and the sensing element which is usually a thermocouple junction or a resistance temperature detector. Models of such cylinders have been dev~loped and are briefly described. The thermal response of these cylinders depends on a number of factors. These include physical properties and dimensions which can be influenced by manufacturing tolerances. Some theoretical and experimental results are presented to exemplify this. Simple representation of complex cylinders is briefly discussed, Some results of thermal processing of food cans are also presented.
Introduction Composite cylinders are not u n c o m o n in present day technology. Besides insulated pipes, the following may be cited as examples: nuclear fuel rods, optical fibers, heat pipes and food cans. A somewhat less obvious example is an industrial thermometer. The components of a composite cylinder attempt to provide one or more functions. The cladding of an optical fiber has a refractive index lower than the core which results in the confinement of a light ray in the core,
*Present address:
Dept. of Chemical Engineering, Technlon City, Haifa 32000, Israel. 417
418
R.K. Chohan
Vol. 14, No. 4
Food cans provide a containing medium as well as a means of preserving A classification the possible designs. (I)
of composite cylinders may assist in conceptualising Three classes of composite cylinders can be identified:
Composite cylinder with annular regions. is composed of different materials
In this class, the cylinder
in annular regions.
One or more of
these annular regions can itself have a composite characteristic perhaps a complex geometric (II)
:food,
form.
Composite cylinders with multiple filaments running longitudinally
with
(or smaller cylinders)
along the cylinder.
(III) Composite cylinders with disperse medium.
In this class, a geometrically
dominant disperse and continuous phase can be identified. Examples of class thermometers.
(I) are insulated pipes and some forms of industrial
Thermocouple
cable leads and some types of optical fiber cables
belong to class (II). Food cans are examples of class
(III).
Not all com-
posite cylinders can neatly be classified as one of the above. some forms of industrial thermometers
Examples are
and optical fiber cables.
This paper attempts to present some of the author's experiences transfer in composite cylinders. belong to class
These include industrial
(I) or a cross between class
of the following discussion
is on the former.
specifically
for industrial thermometers
thermometers,
which
(I) and (II), and food cans. Most Some theoretical and experiment-
al results of the thermal response are presented.
just sensors)
in heat
Though these are obtained
(also called temperature
and food cans, such results should find relevance
sensors or
to other
composites. Factors Affecting Industrial
the Thermal Response
thermometers are composite cylinders which belong to class
(I) or have characteristics
of class (I) and class
of these are shown in figures 1 and 2.
(If). Examples of designs
Further examples are presented
[1,2].
A typical sensor has at least the following components:
sheath,
(ii) an inner layer
(platinum element,
(a ceramic insulant),
in
(i) an outer
(iii) the sensing element
thermistor bead or thermocouple junction which normally
occupies a negligible volume),(iv)
core
present such as in figure 2. Undesired may be introduced during manufacture,
(or mandrel), Additional layers may be components
(such as air or cracks)
handling or use.
Vol. 14, NO. 4
THEI~9~RESPONSE OF O O M P ( ~ I T E ~
419
Processes of interest in composite cylinder analysis and design fall into two categories (3): (a) Short-term processes such as heat transfer within cylinder and between cylinder surface and the environment, flow of fluid surrounding cylinder, vibration, etc. (b) Long-term processes such as corrosion, fatigue, crack and crack propagation, scale formation, erosion, wear, etc. Design considerations such as system function and reliability involves both the above categories. The thermal response of composite cylinders may be influenced by both these categories. The thermal response of temperature sensors is affected by a number of factors [1,5]. Generally, the major determinants of the thermal response are the dimensions and geometrical form of the constituents, physical properties, temperature and pressure. Because of the relatively small outer diameters of average temperature sensors, manufacturing tolerances can be expected to have an influence on the thermal response. It has been found that some of the manufacturing tolerances may have a significant, possibly a dominant, influence [1,2,4].
Some results
presented here exemplify this. Thermal response of systems may be represented as a time history to a specified perturbation or by a single parameter, T .
The latter is usually
taken to be the time required for the response to reach 63.2 percent of the total change following a step perturbation. This defines the time constant which however, strictly, applies to a first-order lumped-parameter system whilst temperature sensors are higher order systems. In fact they are distributed-parameter systems. However, the time constant as a measure of thermal response is widely used. Two kinds of perturbations are of interest for composite cylinders: (a) heat input from the environment, (b) heat input to a location inside the cylinder. Industrial thermometry experimentation and modelling simulates these perturbations as follows: (i)
Plunge test (abbreviation PT) - where the thermometer is plunged into hot water from ambient air,
420
R.K. Chohan
(ii) Electrical
Vol. 14, No. 4
transient test (abbreviated ET) - where the thermometer
is held at a constant temperature water bath at room temperature and the current input into the thermometer approximately
is a step change from
1-5 mA to about 30-50 mA. Substantial
internal
(self) heating results. Most of the results presented below pertain to the first of these two perturbations.
Modeling Attempts of modeling of industrial thermometers have been made. Both distributed
and lumped parameter models have been developed
[5,6]. Some
results are presented here. The distributed-parameters
representation
of a composite medium is
desirable because of the accurate representation process.
of the heat transfer
The finite element method is particularly
suited to composites
because of the relative ease with which differing physical properties be accommodated
In the distributed models,
it was assumed that only radial heat
transfer is present and that the physical properties are constant. temperature
sensor together with a thermowell,
into the following regions: (iv) air gap (inner), sensing element decomposed
can
[14].
(i) thermowell,
(v) aluminum cover,
if present,
(ii) air gap,
The
can be decomposed (iii) sheath,
(vi) ceramic former containing
the
(here a platinum coil). The last region was further
into (a) an outer
wire (assumed to be massless)
(annular) ceramic, imbedded in it,
(b) air gap with sensing
(c) inner cylindrical core.
Other sensor types can be similarly decomposed. Each of these regions four-noded axisymmetric kp and heat capacity, if the thermowell
c.
(steel well, air gap, etc.) was divided into one
line element with appropriate At the thermowell surface
is not present),
appropriate value of
h
thermal conductivity,
(or at the sheath surface
a surface element was added with an
(the surface heat transfer coefficient).
The result-
ing model had 9 elements with 25 nodes, with node 5 from the cylinder center being the temperature of the sensing element. The model was simulated using a finite element package on a minicomputer [5].
Typical outer (inner) diameters of well and sheath were i0 m~ (6,2 mm)
and 6.0 mm (5.07 mm); details can be found in (5) and (8). Figure 3 shows
Vol. 14, No. 4
THEI~MAL RESPONSE (~ CQMIK~ITE CYLINDERS
~
Sheath
~
insulant
Fig. i. A typical sensor for Temperature measurement with possible locations of sensing element: (A) insulated junction, (B) grounded junction, (C) resistance detector.
well heath luminium
cover
ceramic
~ i r /
-sensing
element
gaps Fig. 2. An industrial resistance temperature sensor in a well (cross-sectional view).
0
.
0
Z Ttm
II (secaeds)
~ 14 10
Fig. 3. Comparison between distributed model and experiment.
ZO
421
422
R.K. Chohan
Vol. 14, No. 4
typical model simulation results and experimental data for both types of perturbations described in the last section. The results are for an industrial thermometer
(shown schematically in figure 2). Agreement is seen
to be a good showing the validity of the distributed parameter model. Note that there are two sets of experiments which are distinct and validity is tested with respect to both. The distributed parameter models require a relatively large computer time and memory. Lumped and reduced-order models are desirable in this respect.
It has been found that valid reduced-order models can be constructed
for a class of thermometers shown in figure (2) [7,5].
These validated
models represent the sensor as a third-order lumped parameter system and the sensor-well combination as a fourth-order system. That is, a sensor response can be described by:
T(t)=l+ale-Plt+a2e-P2t+a3e-P3t where the a i , Pi
are constants, T
(1)
the sensor response, t
Two-dimensional lumped models have also been developed
is time. [6].
These
allow radial and axial heat transfer. Except for short cylinders, radial heat transfer is the dominant mode. Two-dimensional models allow study of short cylinders and steam loss errors.
Manufacturin $ Tolerances Manufacturing tolerances appear as dimensional tolerances and physical property variations. Two important tolerances in industrial thermometry are air gap dimensions
(see figure 2) and the porosity of the insulant
(such
as magnesia and alumina). An industrial sensor in a thermowell
(figure 2) was simulated using
the finite element model (described in the last section).
Manufacturing
tolerances in air gap dimensions usually involve very small dimensional changes about a nominal value.
Figure 4 is a typical set of results showing
the effects of dimensional tolerances in the air gaps. For the relatively small changes involved, an almost linear variation in the response time is obtained. A significant variation in the time constants
(50% or more) can
be expected in the manufacture of sensors with thermowells. This sort of variation has been observed in the laboratory 13].
VOl. 14, NO. 4
T}IER~%L RESPONSE (F OOMPOSITE CYLINDERS
Experimental different
results have been obtained recently on the response of
sensors in five liquids.
of manufacturing
423
Some of the results exhibit the effects
tolerances not involving air gaps. These are shown in
figure 5 for two sensors in carbon tetrachloride.
Both the sensors are of
nominal outer diameter 6.20 mm, same make, type and manufacturer. significant difference difference
The
in the thermal response is probably due to the
in the insulant porosity.
First-Order Representation
of a Composite Cylinder
Composite cylinders are distributed parameter systems. Therefore, complete representation
a
of its thermal response would require an infinite
number of modes. As stated earlier, valid third-order lumped models could be constructed
for sensors. Earlier experiences
(5) appear to suggest that
there is one dominant mode, i.e. a sensor may approximately be described by a first-order representation. homogeneous
cylinder,
Simple analysis,
assuming a lumped or
suggests that the time constant may be expressed as [9]: C2 = cI +~-
where
CI
and
C2
(2)
are dependent only on the sensor internal dimensions
and physical properties.
The effects of the external fluid are contained
entirely in the heat transfer coefficient,
h .
Recent experiences also support the approximate validity of equation
(2) [6,10j.
Two composite cylinders of the type shown in figure i
of outer diameters 3 mm and 6 mm with sheath thickness 0.5 ~n and length 20 mm~ were simulated using a two-dimensional model described
in
6 .
The
insulant was alumina. The average response at the center of the cylinder was determined as a function of the heat transfer coefficient.
The results are
presented in figure 6.
i/h
suggesting
A linear relation between
the validity of (2).
T
and
is seen
It should be pointed out that this does
not mean that a first-order representation would agree with the entire time history as shown in figure 3. certain thermocouples,
Other composite cylinders,
were also simulated.
representing
Results were similar to those in
figure 6, again suggesting the approximate validity of expression
(2).
Food Cans Food cans belong to class
(III) composites.
the low length to diameter ratio.
They are characterised by
Significant axial, besides radial, heat
424
R.K. Chohan
Vol.
70
60
50 f f~
40 / /
o
/
30 E
/
J
/ 20
h
f6
26
air gap width (mm x 10 .2 ) Fig. Effect of air gap widths of a temperature detector. and broken line- inner
4. on the thermal response Full line-outer air gap, air gap (see Fig. 2).
8.0
6.0 v
0
~
4.0
$ I,--
2.0
OtS
1:0
Fluid Velocity (m/s) Fig. 5. Thermal response of two sensors of the same type, make, vendor model, and nominal dimension in carbon tetrachloride.
14, No. 4
VOI. 14, No. 4
T ~
~-_~SE
OF C X ~ I T ~
C
~
<=. : " C3 U')
V- l= ~
x~ x
%.60
x
o'.os
6.~5
o'.,o
(M--2.C/W)
1/'H
"~.2o
"lO"
Fig. 6. Time constant-Heat transfer Coefficient relation for a 3 mm (crosses) and 6 mm (squares) sensor.
:--..-;-~'~-- ~
2,0
e • m
' ~
#
/ 4
I
/
^F
C~R
• f v Z00
[
£
;
;
O~ CRUS~ CO~N cAs
~rr~
' I •
. . . . . . . .
SURFACE
[~',
.
.
.
.
...... x,.,
SURFACE OF W~OLECORN CAN * . . . . •
•
..."
p,___._-o
~
I,;
is
Tfeem ( l l n u ~ e s )
Fig. 7. Surface and interior temperatures of crushed and whole corn cans in a still vertical retort.
425
426
R.K. Chohan
Vol. 14, No. 4
transfer may be expected. A number of experiments were performed The cans were steam heated in a vertical and surface measurements
!111,12].
still retort under pressure.
were obtained and are described
Internal
in [11,12]. Complex
temperature histories are possible at internal locations of the cans. Some comparisons
between experimental data are presented
in figure 7.
~his shows the results from two cans, one with 350 g. crushed corn and the other with 350 g. whole corn. Both cans were subjected to similar perturbation (steam heating to 250°F from about 70°F in the retort at about 2 atmosphere absolute pressure). temperature
Observations
can be stated as follows:
(a) the surface
of the whole corn responds faster than that of the crushed corn;
(b) the bottom inside temperature history exhibits a spike for crushed corn, which is absent for whole corn; can was
(c) the center temperature of the whole corn
(not shown in figure 7) much faster than that of the crushed corn.
Other results were obtained for cans containing potato chips in brine and starch solutions. (gelatinisation)
The latter were subject to phase transformation
[12]. This complicates
the heat transfer considerably.
Conclusion Some experience Finite-element
in heat transfer in composite cylinders was presented.
distributed
and lumped models have been developed and some
results were briefly discussed. ing tolerances.
Temperature
sensors are subject to manufactur-
These may have a significant effect on their thermal response.
Large temperature changes may result in physical property variations through its temperature dependence and also through geometrical expansion).
(differential
Model simulations can shed more light on thermometer behaviour
subjected to significant on food
changes
temperature changes
[13]. Some experimental
cans in a vertical still retort were presented.
results
These exemplify the
complex temperature histories that are possible in such cylinders.
References !.
T.W. Kerlin and R.K. Chohan, Chapter 15 in "Handbook of Applications of Measuring Instruments-Electrical Instruments", ed. R.L. Moore, Wiley, to be published (1986).
2.
R.K. Chohan and T.W. Kerlin, Paper 20 in the Proceedings of the Industrial Temperature Measurement Conf., Knoxville, TN (September 1984).
3.
R.K. Chohan and F. Abdullah, "Mathematical Modelling of Thermal Transducers' Progress report, Dept. of Systems Science, The City University, London, (August 1980).
Vol. 14, No. 4
THERMAL R E S P O N S E ~ O O M P O S I T E C Y L I N D E R S
427
4.
RoK, Chohan, "Industrial Thermometry-Dynamic Responsep Usage Environment and Sensor Location"p Submitted A.S.M.E. (1986).
5.
R.K, Chohan, Ph.D. Thesisp The City University, London (1983).
6.
R.K. Chohan, M. Natour and T.W. Kerlin, Paper ii in the Proceedings of the Industrial Temperature Measurement Conference, Knoxville, TN (1984).
7.
R.K. Chohan and F. Abdullah, "Understanding the Dynamic Response of Industrial Temperature Sensors- an Approach Based on Mathematical Models , presented at Transdueer/Tempcon Conf., Earls Court, London,(1983).
8.
R.K. Chohan, F. Abdullah and L. Finkelstein, "Mathematical Modelling of Industrial Thermometers", to be published in Trans. Inst. of Measurement and Control, (1986).
9.
T.W. Kerlin and R. Shepard, "Industrial Temperature Measurement", (1982).
ISA
i0.
T.W. Kerlin i R.K. Chohan and M. Natour, "Effects of Fluid Physical Properties and Flow on the Response of Temperature Sensors", to be submitted (1986).
ii.
J. Mount and R.K. Chohan, "Thermal Processing of Food Cans in a Still Retort", in preparation (1986).
12.
R.K. Chohan and J. Mount, "Simultaneous Surface and Internal Temperature Measurements in Food Cans in Retorts",in preparation (1985).
13.
R.K. Chohan, "The Response of Temperature Sensors", in progress (1986).
14.
L.J. Segerlind, "Applied Finite Element Analysis", Wiley, New York (1976).